id sid tid token lemma pos cana-3688 1 1 microsoft microsoft PROPN cana-3688 1 2 word word PROPN cana-3688 1 3 communications communication NOUN cana-3688 1 4 on on ADP cana-3688 1 5 applied apply VERB cana-3688 1 6 nonlinear nonlinear NOUN cana-3688 1 7 analysis_pro analysis_pro PUNCT cana-3688 1 8 communications communication NOUN cana-3688 1 9 on on ADP cana-3688 1 10 applied apply VERB cana-3688 1 11 nonlinear nonlinear ADJ cana-3688 1 12 analysis analysis NOUN cana-3688 1 13 issn issn NOUN cana-3688 1 14 : : PUNCT cana-3688 1 15 1074 1074 NUM cana-3688 1 16 - - PUNCT cana-3688 1 17 133x 133x NUM cana-3688 1 18 vol vol NOUN cana-3688 1 19 . . PROPN cana-3688 2 1 32 32 NUM cana-3688 2 2 no no INTJ cana-3688 2 3 . . PUNCT cana-3688 3 1 8s 8s PROPN cana-3688 3 2 ( ( PUNCT cana-3688 3 3 2025 2025 NUM cana-3688 3 4 ) ) PUNCT cana-3688 3 5 memory memory NOUN cana-3688 3 6 effect effect NOUN cana-3688 3 7 of of ADP cana-3688 3 8 deformable deformable ADJ cana-3688 3 9 mean mean ADJ cana-3688 3 10 and and CCONJ cana-3688 3 11 deformable deformable ADJ cana-3688 3 12 standard standard ADJ cana-3688 3 13 deviation deviation NOUN cana-3688 3 14 hari hari NOUN cana-3688 3 15 pratap1 pratap1 NOUN cana-3688 3 16 , , PUNCT cana-3688 3 17 sunit sunit PROPN cana-3688 3 18 kumar2 kumar2 PROPN cana-3688 3 19 * * PROPN cana-3688 3 20 , , PUNCT cana-3688 3 21 subodh subodh PROPN cana-3688 3 22 kumar3 kumar3 PROPN cana-3688 3 23 , , PUNCT cana-3688 3 24 gajraj gajraj PROPN cana-3688 3 25 singh4 singh4 PROPN cana-3688 3 26 , , PUNCT cana-3688 3 27 ved ve VERB cana-3688 3 28 pal pal ADJ cana-3688 3 29 singh5 singh5 PROPN cana-3688 4 1 1department 1department NUM cana-3688 4 2 of of ADP cana-3688 4 3 mathematics mathematic NOUN cana-3688 4 4 , , PUNCT cana-3688 4 5 p.g.d.a.v p.g.d.a.v NOUN cana-3688 4 6 . . PUNCT cana-3688 5 1 college college NOUN cana-3688 5 2 evening evening NOUN cana-3688 5 3 , , PUNCT cana-3688 5 4 university university NOUN cana-3688 5 5 of of ADP cana-3688 5 6 delhi-110065 delhi-110065 NOUN cana-3688 5 7 haripratap@pgdave.du.ac.in haripratap@pgdave.du.ac.in PROPN cana-3688 5 8 2department 2department NUM cana-3688 5 9 of of ADP cana-3688 5 10 mathematics mathematic NOUN cana-3688 5 11 , , PUNCT cana-3688 5 12 motilal motilal PROPN cana-3688 5 13 nehru nehru PROPN cana-3688 5 14 college college PROPN cana-3688 5 15 , , PUNCT cana-3688 5 16 university university NOUN cana-3688 5 17 of of ADP cana-3688 5 18 delhi-110021 delhi-110021 NOUN cana-3688 5 19 sunit@mln.du.ac.in sunit@mln.du.ac.in PROPN cana-3688 5 20 3department 3department NUM cana-3688 5 21 of of ADP cana-3688 5 22 mathematics mathematic NOUN cana-3688 5 23 , , PUNCT cana-3688 5 24 shyam shyam PROPN cana-3688 5 25 lal lal PROPN cana-3688 5 26 college college PROPN cana-3688 5 27 , , PUNCT cana-3688 5 28 university university NOUN cana-3688 5 29 of of ADP cana-3688 5 30 delhi-110032 delhi-110032 NOUN cana-3688 6 1 skumarmath@shyamlal.du.ac.in skumarmath@shyamlal.du.ac.in PROPN cana-3688 6 2 4discipline 4discipline NUM cana-3688 6 3 of of ADP cana-3688 6 4 statistics statistic NOUN cana-3688 6 5 , , PUNCT cana-3688 6 6 school school NOUN cana-3688 6 7 of of ADP cana-3688 6 8 sciences science NOUN cana-3688 6 9 , , PUNCT cana-3688 6 10 indira indira PROPN cana-3688 6 11 gandhi gandhi PROPN cana-3688 6 12 national national PROPN cana-3688 6 13 open open PROPN cana-3688 6 14 university university PROPN cana-3688 6 15 , , PUNCT cana-3688 6 16 delhi delhi PROPN cana-3688 6 17 gajrajsingh@ignou.ac.in gajrajsingh@ignou.ac.in PROPN cana-3688 6 18 5department 5department NUM cana-3688 6 19 of of ADP cana-3688 6 20 mathematics mathematic NOUN cana-3688 6 21 , , PUNCT cana-3688 6 22 maharaja maharaja PROPN cana-3688 6 23 agersen agersen PROPN cana-3688 6 24 college college PROPN cana-3688 6 25 , , PUNCT cana-3688 6 26 university university NOUN cana-3688 6 27 of of ADP cana-3688 6 28 delhi-110096 delhi-110096 NOUN cana-3688 6 29 vedpalsingh@mac.du.ac.in vedpalsingh@mac.du.ac.in PROPN cana-3688 6 30 * * PUNCT cana-3688 6 31 corresponding correspond VERB cana-3688 6 32 author author NOUN cana-3688 6 33 : : PUNCT cana-3688 6 34 sunit sunit PROPN cana-3688 6 35 kumar kumar PROPN cana-3688 6 36 article article PROPN cana-3688 6 37 history history NOUN cana-3688 6 38 : : PUNCT cana-3688 6 39 received receive VERB cana-3688 6 40 : : PUNCT cana-3688 6 41 30 30 NUM cana-3688 6 42 - - SYM cana-3688 6 43 10 10 NUM cana-3688 6 44 - - PUNCT cana-3688 6 45 2024 2024 NUM cana-3688 6 46 revised revise VERB cana-3688 6 47 : : PUNCT cana-3688 6 48 04 04 NUM cana-3688 6 49 - - SYM cana-3688 6 50 12 12 NUM cana-3688 6 51 - - PUNCT cana-3688 6 52 2024 2024 NUM cana-3688 6 53 accepted accept VERB cana-3688 6 54 : : PUNCT cana-3688 6 55 27 27 NUM cana-3688 6 56 - - SYM cana-3688 6 57 12 12 NUM cana-3688 6 58 - - PUNCT cana-3688 6 59 2024 2024 NUM cana-3688 6 60 abstract abstract NOUN cana-3688 6 61 : : PUNCT cana-3688 6 62 memory memory NOUN cana-3688 6 63 effects effect NOUN cana-3688 6 64 can can AUX cana-3688 6 65 be be AUX cana-3688 6 66 easily easily ADV cana-3688 6 67 understood understand VERB cana-3688 6 68 with with ADP cana-3688 6 69 the the DET cana-3688 6 70 help help NOUN cana-3688 6 71 of of ADP cana-3688 6 72 fractional fractional ADJ cana-3688 6 73 derivatives derivative NOUN cana-3688 6 74 . . PUNCT cana-3688 7 1 in in ADP cana-3688 7 2 this this DET cana-3688 7 3 paper paper NOUN cana-3688 7 4 with with ADP cana-3688 7 5 the the DET cana-3688 7 6 help help NOUN cana-3688 7 7 of of ADP cana-3688 7 8 deformable deformable ADJ cana-3688 7 9 fractional fractional ADJ cana-3688 7 10 derivative derivative NOUN cana-3688 7 11 we we PRON cana-3688 7 12 try try VERB cana-3688 7 13 to to PART cana-3688 7 14 understand understand VERB cana-3688 7 15 the the DET cana-3688 7 16 role role NOUN cana-3688 7 17 of of ADP cana-3688 7 18 deformability deformability NOUN cana-3688 7 19 on on ADP cana-3688 7 20 memory memory NOUN cana-3688 7 21 effects effect NOUN cana-3688 7 22 of of ADP cana-3688 7 23 a a DET cana-3688 7 24 normal normal ADJ cana-3688 7 25 distribution distribution NOUN cana-3688 7 26 using use VERB cana-3688 7 27 moment moment NOUN cana-3688 7 28 generating generate VERB cana-3688 7 29 function function NOUN cana-3688 7 30 . . PUNCT cana-3688 8 1 first first ADV cana-3688 8 2 , , PUNCT cana-3688 8 3 we we PRON cana-3688 8 4 develop develop VERB cana-3688 8 5 a a DET cana-3688 8 6 formula formula NOUN cana-3688 8 7 for for ADP cana-3688 8 8 deformable deformable ADJ cana-3688 8 9 mean mean ADJ cana-3688 8 10 and and CCONJ cana-3688 8 11 deformable deformable ADJ cana-3688 8 12 standard standard ADJ cana-3688 8 13 deviation deviation NOUN cana-3688 8 14 with with ADP cana-3688 8 15 the the DET cana-3688 8 16 help help NOUN cana-3688 8 17 of of ADP cana-3688 8 18 deformable deformable ADJ cana-3688 8 19 fractional fractional ADJ cana-3688 8 20 derivative derivative NOUN cana-3688 8 21 . . PUNCT cana-3688 9 1 secondly secondly ADV cana-3688 9 2 , , PUNCT cana-3688 9 3 we we PRON cana-3688 9 4 calculate calculate VERB cana-3688 9 5 the the DET cana-3688 9 6 value value NOUN cana-3688 9 7 of of ADP cana-3688 9 8 deformable deformable ADJ cana-3688 9 9 mean mean NOUN cana-3688 9 10 and and CCONJ cana-3688 9 11 deformable deformable ADJ cana-3688 9 12 standard standard ADJ cana-3688 9 13 deviation deviation NOUN cana-3688 9 14 for for ADP cana-3688 9 15 varying vary VERB cana-3688 9 16 fractional fractional ADJ cana-3688 9 17 order order NOUN cana-3688 9 18 . . PUNCT cana-3688 10 1 lastly lastly ADV cana-3688 10 2 , , PUNCT cana-3688 10 3 we we PRON cana-3688 10 4 draw draw VERB cana-3688 10 5 plot plot NOUN cana-3688 10 6 of of ADP cana-3688 10 7 normal normal ADJ cana-3688 10 8 distribution distribution NOUN cana-3688 10 9 for for ADP cana-3688 10 10 calculated calculate VERB cana-3688 10 11 value value NOUN cana-3688 10 12 of of ADP cana-3688 10 13 deformable deformable ADJ cana-3688 10 14 mean mean NOUN cana-3688 10 15 and and CCONJ cana-3688 10 16 deformable deformable ADJ cana-3688 10 17 deviation deviation NOUN cana-3688 10 18 with with ADP cana-3688 10 19 varying vary VERB cana-3688 10 20 fractional fractional ADJ cana-3688 10 21 order order NOUN cana-3688 10 22 and and CCONJ cana-3688 10 23 give give VERB cana-3688 10 24 conclusion conclusion NOUN cana-3688 10 25 . . PUNCT cana-3688 11 1 keywords keyword NOUN cana-3688 11 2 : : PUNCT cana-3688 11 3 moment moment NOUN cana-3688 11 4 generating generate VERB cana-3688 11 5 function function NOUN cana-3688 11 6 , , PUNCT cana-3688 11 7 fractional fractional ADJ cana-3688 11 8 order order NOUN cana-3688 11 9 , , PUNCT cana-3688 11 10 deformable deformable ADJ cana-3688 11 11 derivative derivative ADJ cana-3688 11 12 , , PUNCT cana-3688 11 13 deformable deformable ADJ cana-3688 11 14 mean mean NOUN cana-3688 11 15 and and CCONJ cana-3688 11 16 deformable deformable ADJ cana-3688 11 17 standard standard ADJ cana-3688 11 18 deviation deviation NOUN cana-3688 11 19 . . PUNCT cana-3688 12 1 1 1 X cana-3688 12 2 . . X cana-3688 12 3 introduction introduction NOUN cana-3688 12 4 on on ADP cana-3688 12 5 the the DET cana-3688 12 6 basis basis NOUN cana-3688 12 7 of of ADP cana-3688 12 8 deformable deformable ADJ cana-3688 12 9 fractional fractional ADJ cana-3688 12 10 derivative derivative NOUN cana-3688 12 11 , , PUNCT cana-3688 12 12 we we PRON cana-3688 12 13 will will AUX cana-3688 12 14 define define VERB cana-3688 12 15 deformable deformable ADJ cana-3688 12 16 mean mean ADJ cana-3688 12 17 and and CCONJ cana-3688 12 18 deformable deformable ADJ cana-3688 12 19 standard standard ADJ cana-3688 12 20 deviation deviation NOUN cana-3688 12 21 . . PUNCT cana-3688 13 1 deformable deformable ADJ cana-3688 13 2 fractional fractional ADJ cana-3688 13 3 derivative derivative NOUN cana-3688 13 4 : : PUNCT cana-3688 13 5 let let VERB cana-3688 13 6 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-3688 13 7 ) ) PUNCT cana-3688 13 8 be be AUX cana-3688 13 9 a a DET cana-3688 13 10 real real ADV cana-3688 13 11 valued value VERB cana-3688 13 12 function function NOUN cana-3688 13 13 defined define VERB cana-3688 13 14 on on ADP cana-3688 13 15 interval interval NOUN cana-3688 13 16 ( ( PUNCT cana-3688 13 17 π‘Ž π‘Ž X cana-3688 13 18 , , PUNCT cana-3688 13 19 𝑏 𝑏 NOUN cana-3688 13 20 ) ) PUNCT cana-3688 13 21 for for ADP cana-3688 13 22 a a DET cana-3688 13 23 given give VERB cana-3688 13 24 number number NOUN cana-3688 13 25 𝛼 𝛼 NOUN cana-3688 13 26 , , PUNCT cana-3688 14 1 0 0 NUM cana-3688 14 2 ≀ ≀ NUM cana-3688 15 1 𝛼 𝛼 X cana-3688 15 2 ≀ ≀ NUM cana-3688 15 3 1 1 NUM cana-3688 15 4 lim lim NOUN cana-3688 15 5 β†’ β†’ PUNCT cana-3688 15 6 ( ( PUNCT cana-3688 15 7 ) ) PUNCT cana-3688 15 8 ( ( PUNCT cana-3688 15 9 ) ) PUNCT cana-3688 15 10 ( ( PUNCT cana-3688 15 11 ) ) PUNCT cana-3688 15 12 ( ( PUNCT cana-3688 15 13 1 1 X cana-3688 15 14 ) ) PUNCT cana-3688 15 15 where where SCONJ cana-3688 15 16 𝛼 𝛼 X cana-3688 15 17 + + NOUN cana-3688 15 18 𝛽 𝛽 NOUN cana-3688 15 19 = = SYM cana-3688 15 20 1 1 NUM cana-3688 15 21 if if SCONJ cana-3688 15 22 this this DET cana-3688 15 23 limit limit NOUN cana-3688 15 24 exists exist VERB cana-3688 15 25 , , PUNCT cana-3688 15 26 we we PRON cana-3688 15 27 denote denote VERB cana-3688 15 28 it it PRON cana-3688 15 29 by by ADP cana-3688 15 30 𝐷 𝐷 NOUN cana-3688 15 31 [ [ NOUN cana-3688 15 32 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-3688 15 33 ) ) PUNCT cana-3688 15 34 ] ] PUNCT cana-3688 16 1 [ [ X cana-3688 16 2 1 1 NUM cana-3688 16 3 ] ] PUNCT cana-3688 16 4 . . PUNCT cana-3688 17 1 𝐷 𝐷 NOUN cana-3688 17 2 [ [ NOUN cana-3688 17 3 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-3688 17 4 ) ) PUNCT cana-3688 17 5 ] ] PUNCT cana-3688 18 1 = = PUNCT cana-3688 18 2 𝛼 𝛼 X cana-3688 18 3 𝐷𝑓(π‘₯ 𝐷𝑓(π‘₯ NOUN cana-3688 18 4 ) ) PUNCT cana-3688 18 5 + + NUM cana-3688 18 6 𝛽𝑓(π‘₯ 𝛽𝑓(π‘₯ NOUN cana-3688 18 7 ) ) PUNCT cana-3688 18 8 ( ( PUNCT cana-3688 18 9 2 2 X cana-3688 18 10 ) ) PUNCT cana-3688 18 11 or or CCONJ cana-3688 18 12 𝐷 𝐷 NOUN cana-3688 18 13 [ [ NOUN cana-3688 18 14 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-3688 18 15 ) ) PUNCT cana-3688 18 16 ] ] PUNCT cana-3688 19 1 = = PUNCT cana-3688 19 2 𝛼 𝛼 X cana-3688 19 3 𝐷𝑓(π‘₯ 𝐷𝑓(π‘₯ NOUN cana-3688 19 4 ) ) PUNCT cana-3688 20 1 + + CCONJ cana-3688 20 2 ( ( PUNCT cana-3688 20 3 1 1 NUM cana-3688 20 4 βˆ’ βˆ’ NOUN cana-3688 20 5 𝛼 𝛼 NOUN cana-3688 20 6 ) ) PUNCT cana-3688 20 7 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-3688 20 8 ) ) PUNCT cana-3688 20 9 https://internationalpubls.com https://internationalpubls.com X cana-3688 20 10 439 439 NUM cana-3688 20 11 communications communication NOUN cana-3688 20 12 on on ADP cana-3688 20 13 applied apply VERB cana-3688 20 14 nonlinear nonlinear ADJ cana-3688 20 15 analysis analysis NOUN cana-3688 20 16 issn issn NOUN cana-3688 20 17 : : PUNCT cana-3688 20 18 1074 1074 NUM cana-3688 20 19 - - PUNCT cana-3688 20 20 133x 133x NUM cana-3688 20 21 vol vol NOUN cana-3688 20 22 . . PROPN cana-3688 21 1 32 32 NUM cana-3688 21 2 no no INTJ cana-3688 21 3 . . PUNCT cana-3688 22 1 8s 8s PROPN cana-3688 22 2 ( ( PUNCT cana-3688 22 3 2025 2025 NUM cana-3688 22 4 ) ) PUNCT cana-3688 22 5 moment moment NOUN cana-3688 22 6 generating generate VERB cana-3688 22 7 function function NOUN cana-3688 22 8 : : PUNCT cana-3688 22 9 let let VERB cana-3688 22 10 𝑋 𝑋 NOUN cana-3688 22 11 be be AUX cana-3688 22 12 random random ADJ cana-3688 22 13 variable variable NOUN cana-3688 22 14 ( ( PUNCT cana-3688 22 15 r.v r.v PROPN cana-3688 22 16 . . PROPN cana-3688 22 17 ) ) PUNCT cana-3688 23 1 such such ADJ cana-3688 23 2 that that PRON cana-3688 23 3 for for ADP cana-3688 23 4 some some DET cana-3688 23 5 πœ– πœ– PROPN cana-3688 23 6 > > X cana-3688 23 7 0 0 PROPN cana-3688 23 8 , , PUNCT cana-3688 23 9 the the DET cana-3688 23 10 expected expect VERB cana-3688 23 11 value value NOUN cana-3688 23 12 of of ADP cana-3688 23 13 𝑒 𝑒 PROPN cana-3688 23 14 exists exist VERB cana-3688 23 15 for for ADP cana-3688 23 16 βˆ’πœ– βˆ’πœ– NOUN cana-3688 23 17 < < X cana-3688 23 18 𝑑 𝑑 X cana-3688 23 19 < < X cana-3688 23 20 πœ–. πœ–. NOUN cana-3688 23 21 then then ADV cana-3688 23 22 moment moment VERB cana-3688 23 23 generating generate VERB cana-3688 23 24 function function NOUN cana-3688 23 25 ( ( PUNCT cana-3688 23 26 m.g.f m.g.f PROPN cana-3688 23 27 . . PUNCT cana-3688 23 28 ) ) PUNCT cana-3688 23 29 of of ADP cana-3688 23 30 𝑋 𝑋 PROPN cana-3688 23 31 is be AUX cana-3688 23 32 defined define VERB cana-3688 23 33 to to PART cana-3688 23 34 be be AUX cana-3688 23 35 the the DET cana-3688 23 36 function function NOUN cana-3688 23 37 𝑀 𝑀 PROPN cana-3688 23 38 ( ( PUNCT cana-3688 23 39 𝑑 𝑑 NOUN cana-3688 23 40 ) ) PUNCT cana-3688 23 41 = = SYM cana-3688 23 42 𝐸[𝑒 𝐸[𝑒 X cana-3688 23 43 ] ] X cana-3688 23 44 , , PUNCT cana-3688 23 45 for for ADP cana-3688 23 46 βˆ’πœ– βˆ’πœ– NOUN cana-3688 23 47 < < X cana-3688 23 48 𝑑 𝑑 X cana-3688 23 49 < < X cana-3688 23 50 πœ– πœ– PROPN cana-3688 24 1 [ [ X cana-3688 24 2 3],[5]-[6 3],[5]-[6 NOUN cana-3688 24 3 ] ] PUNCT cana-3688 24 4 . . PUNCT cana-3688 25 1 𝑀 𝑀 PROPN cana-3688 25 2 ( ( PUNCT cana-3688 25 3 𝑑 𝑑 NOUN cana-3688 25 4 ) ) PUNCT cana-3688 25 5 = = SYM cana-3688 25 6 𝐸[𝑒 𝐸[𝑒 X cana-3688 25 7 ] ] PUNCT cana-3688 26 1 = = PUNCT cana-3688 26 2 ∫ ∫ PROPN cana-3688 26 3 𝑒 𝑒 PROPN cana-3688 26 4 𝑓 𝑓 PROPN cana-3688 26 5 ( ( PUNCT cana-3688 26 6 π‘₯)𝑑π‘₯ π‘₯)𝑑π‘₯ PROPN cana-3688 26 7 ( ( PUNCT cana-3688 26 8 3 3 NUM cana-3688 26 9 ) ) PUNCT cana-3688 26 10 where where SCONJ cana-3688 26 11 𝑋 𝑋 NOUN cana-3688 26 12 is be AUX cana-3688 26 13 a a DET cana-3688 26 14 continuous continuous ADJ cana-3688 26 15 r.v r.v NOUN cana-3688 26 16 . . PROPN cana-3688 27 1 and and CCONJ cana-3688 27 2 𝑓 𝑓 DET cana-3688 27 3 ( ( PUNCT cana-3688 27 4 π‘₯ π‘₯ NOUN cana-3688 27 5 ) ) PUNCT cana-3688 27 6 is be AUX cana-3688 27 7 a a DET cana-3688 27 8 probability probability NOUN cana-3688 27 9 density density NOUN cana-3688 27 10 function function NOUN cana-3688 27 11 s.t s.t PROPN cana-3688 27 12 . . PUNCT cana-3688 28 1 [ [ X cana-3688 28 2 3 3 NUM cana-3688 28 3 ] ] PUNCT cana-3688 28 4 , , PUNCT cana-3688 28 5 [ [ X cana-3688 28 6 5 5 NUM cana-3688 28 7 ] ] PUNCT cana-3688 28 8 . . PUNCT cana-3688 29 1 ( ( PUNCT cana-3688 29 2 i i NOUN cana-3688 29 3 ) ) PUNCT cana-3688 29 4 𝑓 𝑓 PROPN cana-3688 29 5 ( ( PUNCT cana-3688 29 6 π‘₯ π‘₯ NOUN cana-3688 29 7 ) ) PUNCT cana-3688 29 8 β‰₯ β‰₯ NOUN cana-3688 29 9 0 0 NUM cana-3688 29 10 ( ( PUNCT cana-3688 29 11 ii ii PROPN cana-3688 29 12 ) ) PUNCT cana-3688 29 13 ∫ ∫ PROPN cana-3688 30 1 𝑓 𝑓 PROPN cana-3688 30 2 ( ( PUNCT cana-3688 30 3 π‘₯)𝑑π‘₯ π‘₯)𝑑π‘₯ PROPN cana-3688 30 4 = = SYM cana-3688 30 5 1 1 NUM cana-3688 30 6 or or CCONJ cana-3688 30 7 𝑀 𝑀 PROPN cana-3688 30 8 ( ( PUNCT cana-3688 30 9 𝑑 𝑑 NOUN cana-3688 30 10 ) ) PUNCT cana-3688 30 11 = = SYM cana-3688 30 12 𝐸[𝑒 𝐸[𝑒 X cana-3688 30 13 ] ] PUNCT cana-3688 31 1 = = PUNCT cana-3688 31 2 𝑒 𝑒 PROPN cana-3688 31 3 𝑝 𝑝 PROPN cana-3688 31 4 ( ( PUNCT cana-3688 31 5 π‘₯ π‘₯ NOUN cana-3688 31 6 ) ) PUNCT cana-3688 31 7 ∈ ∈ NOUN cana-3688 31 8 where where SCONJ cana-3688 31 9 𝑋 𝑋 NOUN cana-3688 31 10 is be AUX cana-3688 31 11 a a DET cana-3688 31 12 discrete discrete ADJ cana-3688 31 13 r.v r.v NOUN cana-3688 31 14 . . PROPN cana-3688 31 15 with with ADP cana-3688 31 16 space space NOUN cana-3688 31 17 Ο‰ Ο‰ PROPN cana-3688 31 18 and and CCONJ cana-3688 31 19 𝑝 𝑝 PROPN cana-3688 31 20 ( ( PUNCT cana-3688 31 21 π‘₯ π‘₯ NOUN cana-3688 31 22 ) ) PUNCT cana-3688 31 23 is be AUX cana-3688 31 24 a a DET cana-3688 31 25 probability probability NOUN cana-3688 31 26 mass mass NOUN cana-3688 31 27 function function NOUN cana-3688 31 28 s.t s.t PROPN cana-3688 31 29 . . PUNCT cana-3688 32 1 [ [ X cana-3688 32 2 3 3 NUM cana-3688 32 3 ] ] PUNCT cana-3688 32 4 , , PUNCT cana-3688 32 5 [ [ X cana-3688 32 6 5 5 NUM cana-3688 32 7 ] ] PUNCT cana-3688 32 8 . . PUNCT cana-3688 33 1 ( ( PUNCT cana-3688 33 2 i i NOUN cana-3688 33 3 ) ) PUNCT cana-3688 33 4 0 0 NUM cana-3688 34 1 ≀ ≀ PROPN cana-3688 34 2 𝑝 𝑝 PROPN cana-3688 34 3 ( ( PUNCT cana-3688 34 4 π‘₯ π‘₯ NOUN cana-3688 34 5 ) ) PUNCT cana-3688 34 6 ≀ ≀ NUM cana-3688 34 7 1 1 NUM cana-3688 34 8 , , PUNCT cana-3688 34 9 π‘₯ π‘₯ PRON cana-3688 34 10 ∈ ∈ PROPN cana-3688 34 11 Ο‰ Ο‰ X cana-3688 34 12 ( ( PUNCT cana-3688 34 13 ii ii PROPN cana-3688 34 14 ) ) PUNCT cana-3688 34 15 βˆ‘ βˆ‘ PROPN cana-3688 34 16 𝑝 𝑝 PROPN cana-3688 34 17 ( ( PUNCT cana-3688 34 18 π‘₯)∈ π‘₯)∈ X cana-3688 34 19 = = SYM cana-3688 34 20 1 1 NUM cana-3688 34 21 2 2 NUM cana-3688 34 22 . . PUNCT cana-3688 34 23 deformable deformable ADJ cana-3688 34 24 mean mean NOUN cana-3688 34 25 if if SCONJ cana-3688 34 26 𝑀 𝑀 PROPN cana-3688 34 27 ( ( PUNCT cana-3688 34 28 𝑑 𝑑 NOUN cana-3688 34 29 ) ) PUNCT cana-3688 34 30 is be AUX cana-3688 34 31 a a DET cana-3688 34 32 well well ADV cana-3688 34 33 - - PUNCT cana-3688 34 34 defined define VERB cana-3688 34 35 moment moment NOUN cana-3688 34 36 generating generate VERB cana-3688 34 37 function function NOUN cana-3688 34 38 then then ADV cana-3688 34 39 we we PRON cana-3688 34 40 can can AUX cana-3688 34 41 define define VERB cana-3688 34 42 deformable deformable ADJ cana-3688 34 43 mean mean NOUN cana-3688 34 44 ( ( PUNCT cana-3688 34 45 πœ‡ πœ‡ X cana-3688 34 46 ) ) PUNCT cana-3688 34 47 with with ADP cana-3688 34 48 the the DET cana-3688 34 49 help help NOUN cana-3688 34 50 of of ADP cana-3688 34 51 moment moment NOUN cana-3688 34 52 generating generate VERB cana-3688 34 53 function function NOUN cana-3688 34 54 as as ADP cana-3688 34 55 πœ‡ πœ‡ ADP cana-3688 34 56 = = X cana-3688 34 57 𝑀 𝑀 PROPN cana-3688 34 58 ( ( PUNCT cana-3688 34 59 0 0 NUM cana-3688 34 60 ) ) PUNCT cana-3688 34 61 deformable deformable ADJ cana-3688 34 62 derivative derivative NOUN cana-3688 34 63 of of ADP cana-3688 34 64 equation equation NOUN cana-3688 34 65 ( ( PUNCT cana-3688 34 66 3 3 NUM cana-3688 34 67 ) ) PUNCT cana-3688 34 68 w.r.to w.r.to NOUN cana-3688 35 1 t t PROPN cana-3688 35 2 𝑀 𝑀 PROPN cana-3688 35 3 ( ( PUNCT cana-3688 35 4 𝑑 𝑑 NOUN cana-3688 35 5 ) ) PUNCT cana-3688 35 6 = = SYM cana-3688 35 7 ∫ ∫ PROPN cana-3688 35 8 𝐷 𝐷 PROPN cana-3688 35 9 [ [ X cana-3688 35 10 𝑒 𝑒 PROPN cana-3688 35 11 ] ] X cana-3688 35 12 𝑓 𝑓 PROPN cana-3688 35 13 ( ( PUNCT cana-3688 35 14 π‘₯)𝑑π‘₯ π‘₯)𝑑π‘₯ PROPN cana-3688 35 15 ( ( PUNCT cana-3688 35 16 4 4 NUM cana-3688 35 17 ) ) PUNCT cana-3688 35 18 𝑀 𝑀 PROPN cana-3688 35 19 ( ( PUNCT cana-3688 35 20 𝑑 𝑑 NOUN cana-3688 35 21 ) ) PUNCT cana-3688 35 22 = = PRON cana-3688 35 23 { { PUNCT cana-3688 35 24 𝛼 𝛼 PART cana-3688 35 25 π‘₯ π‘₯ X cana-3688 35 26 𝑒 𝑒 PROPN cana-3688 35 27 + + CCONJ cana-3688 35 28 ( ( PUNCT cana-3688 35 29 1 1 NUM cana-3688 35 30 βˆ’ βˆ’ NOUN cana-3688 35 31 𝛼 𝛼 NOUN cana-3688 35 32 ) ) PUNCT cana-3688 35 33 𝑒 𝑒 PROPN cana-3688 35 34 } } PUNCT cana-3688 35 35 𝑓 𝑓 PROPN cana-3688 35 36 ( ( PUNCT cana-3688 35 37 π‘₯)𝑑π‘₯ π‘₯)𝑑π‘₯ PROPN cana-3688 35 38 𝑀 𝑀 PROPN cana-3688 35 39 ( ( PUNCT cana-3688 35 40 𝑑 𝑑 NOUN cana-3688 35 41 ) ) PUNCT cana-3688 35 42 = = PUNCT cana-3688 35 43 𝛼 𝛼 NOUN cana-3688 35 44 ∫ ∫ PROPN cana-3688 36 1 π‘₯ π‘₯ X cana-3688 36 2 𝑒 𝑒 PROPN cana-3688 36 3 𝑓 𝑓 PROPN cana-3688 36 4 ( ( PUNCT cana-3688 36 5 π‘₯ π‘₯ NOUN cana-3688 36 6 ) ) PUNCT cana-3688 36 7 𝑑π‘₯ 𝑑π‘₯ VERB cana-3688 36 8 + + X cana-3688 36 9 ( ( PUNCT cana-3688 36 10 1 1 NUM cana-3688 36 11 βˆ’ βˆ’ NOUN cana-3688 36 12 𝛼 𝛼 X cana-3688 36 13 ) ) PUNCT cana-3688 36 14 ∫ ∫ PROPN cana-3688 36 15 𝑒 𝑒 PROPN cana-3688 36 16 𝑓 𝑓 PROPN cana-3688 36 17 ( ( PUNCT cana-3688 36 18 π‘₯)𝑑π‘₯ π‘₯)𝑑π‘₯ PROPN cana-3688 36 19 ( ( PUNCT cana-3688 36 20 5 5 NUM cana-3688 36 21 ) ) PUNCT cana-3688 36 22 𝑀 𝑀 PROPN cana-3688 36 23 ( ( PUNCT cana-3688 36 24 0 0 NUM cana-3688 36 25 ) ) PUNCT cana-3688 36 26 = = PRON cana-3688 37 1 𝛼 𝛼 NOUN cana-3688 37 2 π‘₯ π‘₯ X cana-3688 37 3 𝑒 𝑒 PROPN cana-3688 37 4 𝑓 𝑓 DET cana-3688 37 5 ( ( PUNCT cana-3688 37 6 π‘₯ π‘₯ NOUN cana-3688 37 7 ) ) PUNCT cana-3688 37 8 𝑑π‘₯ 𝑑π‘₯ VERB cana-3688 37 9 + + X cana-3688 37 10 ( ( PUNCT cana-3688 37 11 1 1 NUM cana-3688 37 12 βˆ’ βˆ’ NOUN cana-3688 37 13 𝛼 𝛼 NOUN cana-3688 37 14 ) ) PUNCT cana-3688 37 15 𝑒 𝑒 PROPN cana-3688 37 16 𝑓 𝑓 PROPN cana-3688 37 17 ( ( PUNCT cana-3688 37 18 π‘₯)𝑑π‘₯ π‘₯)𝑑π‘₯ PROPN cana-3688 37 19 𝑀 𝑀 PROPN cana-3688 37 20 ( ( PUNCT cana-3688 37 21 0 0 NUM cana-3688 37 22 ) ) PUNCT cana-3688 37 23 = = PRON cana-3688 37 24 𝛼 𝛼 PART cana-3688 37 25 π‘₯ π‘₯ X cana-3688 37 26 𝑓 𝑓 PROPN cana-3688 37 27 ( ( PUNCT cana-3688 37 28 π‘₯ π‘₯ NOUN cana-3688 37 29 ) ) PUNCT cana-3688 37 30 𝑑π‘₯ 𝑑π‘₯ VERB cana-3688 37 31 + + X cana-3688 37 32 ( ( PUNCT cana-3688 37 33 1 1 NUM cana-3688 37 34 βˆ’ βˆ’ NOUN cana-3688 37 35 𝛼 𝛼 NOUN cana-3688 37 36 ) ) PUNCT cana-3688 37 37 𝑓 𝑓 PROPN cana-3688 37 38 ( ( PUNCT cana-3688 37 39 π‘₯ π‘₯ NOUN cana-3688 37 40 ) ) PUNCT cana-3688 37 41 𝑑π‘₯ 𝑑π‘₯ VERB cana-3688 37 42 𝑀 𝑀 PROPN cana-3688 37 43 ( ( PUNCT cana-3688 37 44 0 0 NUM cana-3688 37 45 ) ) PUNCT cana-3688 37 46 = = PUNCT cana-3688 37 47 𝛼 𝛼 X cana-3688 37 48 𝐸(𝑋 𝐸(𝑋 NOUN cana-3688 37 49 ) ) PUNCT cana-3688 37 50 + + CCONJ cana-3688 38 1 ( ( PUNCT cana-3688 38 2 1 1 NUM cana-3688 38 3 βˆ’ βˆ’ NOUN cana-3688 38 4 𝛼 𝛼 NOUN cana-3688 38 5 ) ) PUNCT cana-3688 38 6 ( ( PUNCT cana-3688 38 7 6 6 NUM cana-3688 38 8 ) ) PUNCT cana-3688 38 9 or or CCONJ cana-3688 38 10 πœ‡ πœ‡ X cana-3688 38 11 = = X cana-3688 38 12 𝛼 𝛼 X cana-3688 38 13 πœ‡ πœ‡ X cana-3688 38 14 + + X cana-3688 38 15 ( ( PUNCT cana-3688 38 16 1 1 NUM cana-3688 38 17 βˆ’ βˆ’ NOUN cana-3688 38 18 𝛼 𝛼 NOUN cana-3688 38 19 ) ) PUNCT cana-3688 38 20 for for ADP cana-3688 38 21 πœ‡ πœ‡ X cana-3688 38 22 ( ( PUNCT cana-3688 38 23 deformable deformable ADJ cana-3688 38 24 mean mean NOUN cana-3688 38 25 ) ) PUNCT cana-3688 38 26 following follow VERB cana-3688 38 27 cases case NOUN cana-3688 38 28 arises arise VERB cana-3688 38 29 : : PUNCT cana-3688 38 30 ( ( PUNCT cana-3688 38 31 i i NOUN cana-3688 38 32 ) ) PUNCT cana-3688 38 33 𝛼 𝛼 NOUN cana-3688 38 34 = = SYM cana-3688 38 35 0 0 NUM cana-3688 38 36 , , PUNCT cana-3688 38 37 and and CCONJ cana-3688 38 38 πœ‡ πœ‡ ADP cana-3688 38 39 ∈ ∈ PROPN cana-3688 38 40 𝑅 𝑅 PROPN cana-3688 38 41 then then ADV cana-3688 38 42 πœ‡ πœ‡ SCONJ cana-3688 38 43 = = NOUN cana-3688 38 44 1 1 NUM cana-3688 38 45 ( ( PUNCT cana-3688 38 46 7 7 NUM cana-3688 38 47 ) ) PUNCT cana-3688 38 48 https://internationalpubls.com https://internationalpubls.com X cana-3688 38 49 440 440 NUM cana-3688 38 50 communications communication NOUN cana-3688 38 51 on on ADP cana-3688 38 52 applied apply VERB cana-3688 38 53 nonlinear nonlinear ADJ cana-3688 38 54 analysis analysis NOUN cana-3688 38 55 issn issn NOUN cana-3688 38 56 : : PUNCT cana-3688 38 57 1074 1074 NUM cana-3688 38 58 - - PUNCT cana-3688 38 59 133x 133x NUM cana-3688 38 60 vol vol NOUN cana-3688 38 61 . . PROPN cana-3688 39 1 32 32 NUM cana-3688 39 2 no no INTJ cana-3688 39 3 . . PUNCT cana-3688 40 1 8s 8s PROPN cana-3688 40 2 ( ( PUNCT cana-3688 40 3 2025 2025 NUM cana-3688 40 4 ) ) PUNCT cana-3688 40 5 ( ( PUNCT cana-3688 40 6 ii ii NOUN cana-3688 40 7 ) ) PUNCT cana-3688 40 8 𝛼 𝛼 PROPN cana-3688 40 9 = = X cana-3688 40 10 , , PUNCT cana-3688 40 11 and and CCONJ cana-3688 40 12 πœ‡ πœ‡ ADP cana-3688 40 13 ∈ ∈ PROPN cana-3688 40 14 𝑅 𝑅 PROPN cana-3688 40 15 then then ADV cana-3688 40 16 πœ‡ πœ‡ ADP cana-3688 40 17 = = X cana-3688 40 18 ( ( PUNCT cana-3688 40 19 πœ‡ πœ‡ X cana-3688 40 20 + + X cana-3688 40 21 1 1 NUM cana-3688 40 22 ) ) PUNCT cana-3688 40 23 ( ( PUNCT cana-3688 40 24 8) 8) NUM cana-3688 40 25 ( ( PUNCT cana-3688 40 26 iii iii NOUN cana-3688 40 27 ) ) PUNCT cana-3688 40 28 𝛼 𝛼 NOUN cana-3688 40 29 = = SYM cana-3688 40 30 1 1 NUM cana-3688 40 31 , , PUNCT cana-3688 40 32 and and CCONJ cana-3688 40 33 πœ‡ πœ‡ ADP cana-3688 40 34 ∈ ∈ PROPN cana-3688 40 35 𝑅 𝑅 PROPN cana-3688 40 36 then then ADV cana-3688 40 37 πœ‡ πœ‡ ADP cana-3688 40 38 = = X cana-3688 40 39 πœ‡ πœ‡ X cana-3688 40 40 ( ( PUNCT cana-3688 40 41 9 9 NUM cana-3688 40 42 ) ) PUNCT cana-3688 40 43 ( ( PUNCT cana-3688 40 44 iv iv X cana-3688 40 45 ) ) PUNCT cana-3688 40 46 0 0 PUNCT cana-3688 41 1 < < X cana-3688 41 2 𝛼 𝛼 X cana-3688 41 3 < < X cana-3688 41 4 1 1 NUM cana-3688 41 5 , , PUNCT cana-3688 41 6 and and CCONJ cana-3688 41 7 πœ‡ πœ‡ ADP cana-3688 41 8 ∈ ∈ PROPN cana-3688 41 9 𝑅 𝑅 PROPN cana-3688 41 10 then then ADV cana-3688 41 11 πœ‡ πœ‡ X cana-3688 41 12 = = X cana-3688 41 13 π›Όπœ‡ π›Όπœ‡ ADJ cana-3688 41 14 + + CCONJ cana-3688 41 15 ( ( PUNCT cana-3688 41 16 1 1 NUM cana-3688 41 17 βˆ’ βˆ’ NOUN cana-3688 41 18 𝛼 𝛼 NOUN cana-3688 41 19 ) ) PUNCT cana-3688 41 20 ( ( PUNCT cana-3688 41 21 10 10 NUM cana-3688 41 22 ) ) PUNCT cana-3688 41 23 3 3 NUM cana-3688 41 24 . . PUNCT cana-3688 41 25 deformable deformable ADJ cana-3688 41 26 standard standard ADJ cana-3688 41 27 deviation deviation NOUN cana-3688 41 28 if if SCONJ cana-3688 41 29 𝑀 𝑀 PROPN cana-3688 41 30 ( ( PUNCT cana-3688 41 31 𝑑 𝑑 NOUN cana-3688 41 32 ) ) PUNCT cana-3688 41 33 is be AUX cana-3688 41 34 a a DET cana-3688 41 35 well well ADV cana-3688 41 36 - - PUNCT cana-3688 41 37 defined define VERB cana-3688 41 38 moment moment NOUN cana-3688 41 39 generating generate VERB cana-3688 41 40 function function NOUN cana-3688 41 41 then then ADV cana-3688 41 42 we we PRON cana-3688 41 43 denote denote VERB cana-3688 41 44 deformable deformable ADJ cana-3688 41 45 variance variance NOUN cana-3688 41 46 by by ADP cana-3688 41 47 ( ( PUNCT cana-3688 41 48 𝜎 𝜎 PROPN cana-3688 41 49 ) ) PUNCT cana-3688 41 50 and and CCONJ cana-3688 41 51 defined define VERB cana-3688 41 52 as as ADP cana-3688 41 53 ( ( PUNCT cana-3688 41 54 𝜎 𝜎 NOUN cana-3688 41 55 ) ) PUNCT cana-3688 41 56 = = SYM cana-3688 41 57 𝑀 𝑀 PROPN cana-3688 41 58 . . PUNCT cana-3688 42 1 ( ( PUNCT cana-3688 42 2 0 0 NUM cana-3688 42 3 ) ) PUNCT cana-3688 42 4 βˆ’ βˆ’ PROPN cana-3688 42 5 ( ( PUNCT cana-3688 42 6 𝑀 𝑀 PROPN cana-3688 42 7 ( ( PUNCT cana-3688 42 8 0 0 NUM cana-3688 42 9 ) ) PUNCT cana-3688 42 10 ) ) PUNCT cana-3688 42 11 ( ( PUNCT cana-3688 42 12 11 11 X cana-3688 42 13 ) ) PUNCT cana-3688 42 14 deformable deformable ADJ cana-3688 42 15 derivative derivative NOUN cana-3688 42 16 of of ADP cana-3688 42 17 equation equation NOUN cana-3688 42 18 ( ( PUNCT cana-3688 42 19 5 5 NUM cana-3688 42 20 ) ) PUNCT cana-3688 42 21 w.r.to w.r.to NOUN cana-3688 42 22 t t PROPN cana-3688 42 23 𝑀 𝑀 PROPN cana-3688 42 24 . . PUNCT cana-3688 43 1 ( ( PUNCT cana-3688 43 2 𝑑 𝑑 X cana-3688 43 3 ) ) PUNCT cana-3688 43 4 = = PUNCT cana-3688 43 5 𝛼 𝛼 NOUN cana-3688 43 6 ∫ ∫ PROPN cana-3688 43 7 π‘₯ π‘₯ PROPN cana-3688 43 8 𝐷 𝐷 PROPN cana-3688 43 9 { { PUNCT cana-3688 43 10 𝑒 𝑒 PROPN cana-3688 43 11 } } PUNCT cana-3688 43 12 𝑓 𝑓 PROPN cana-3688 43 13 ( ( PUNCT cana-3688 43 14 π‘₯ π‘₯ NOUN cana-3688 43 15 ) ) PUNCT cana-3688 43 16 𝑑π‘₯ 𝑑π‘₯ VERB cana-3688 43 17 + + X cana-3688 43 18 ( ( PUNCT cana-3688 43 19 1 1 NUM cana-3688 43 20 βˆ’ βˆ’ NOUN cana-3688 43 21 𝛼 𝛼 X cana-3688 43 22 ) ) PUNCT cana-3688 43 23 ∫ ∫ PROPN cana-3688 43 24 𝐷 𝐷 PROPN cana-3688 43 25 { { PUNCT cana-3688 43 26 𝑒 𝑒 PROPN cana-3688 43 27 } } PUNCT cana-3688 43 28 𝑓 𝑓 PROPN cana-3688 43 29 ( ( PUNCT cana-3688 43 30 π‘₯)𝑑π‘₯ π‘₯)𝑑π‘₯ PROPN cana-3688 43 31 𝑀 𝑀 PROPN cana-3688 43 32 . . PUNCT cana-3688 44 1 ( ( PUNCT cana-3688 44 2 𝑑 𝑑 X cana-3688 44 3 ) ) PUNCT cana-3688 44 4 = = SYM cana-3688 45 1 𝛼 𝛼 PART cana-3688 45 2 π‘₯ π‘₯ X cana-3688 45 3 { { PUNCT cana-3688 45 4 𝛼 𝛼 NOUN cana-3688 45 5 π‘₯ π‘₯ X cana-3688 45 6 𝑒 𝑒 PROPN cana-3688 45 7 + + CCONJ cana-3688 45 8 ( ( PUNCT cana-3688 45 9 1 1 NUM cana-3688 45 10 βˆ’ βˆ’ NOUN cana-3688 45 11 𝛼 𝛼 NOUN cana-3688 45 12 ) ) PUNCT cana-3688 45 13 𝑒 𝑒 PROPN cana-3688 45 14 } } PUNCT cana-3688 45 15 𝑓 𝑓 PROPN cana-3688 45 16 ( ( PUNCT cana-3688 45 17 π‘₯ π‘₯ NOUN cana-3688 45 18 ) ) PUNCT cana-3688 45 19 𝑑π‘₯ 𝑑π‘₯ VERB cana-3688 45 20 + + X cana-3688 45 21 ( ( PUNCT cana-3688 45 22 1 1 NUM cana-3688 45 23 βˆ’ βˆ’ NOUN cana-3688 45 24 𝛼 𝛼 NOUN cana-3688 45 25 ) ) PUNCT cana-3688 45 26 { { PUNCT cana-3688 45 27 𝛼 𝛼 PART cana-3688 45 28 π‘₯ π‘₯ X cana-3688 45 29 𝑒 𝑒 PROPN cana-3688 45 30 + + CCONJ cana-3688 45 31 ( ( PUNCT cana-3688 45 32 1 1 NUM cana-3688 45 33 βˆ’ βˆ’ NOUN cana-3688 45 34 𝛼 𝛼 NOUN cana-3688 45 35 ) ) PUNCT cana-3688 45 36 𝑒 𝑒 PROPN cana-3688 45 37 } } PUNCT cana-3688 45 38 𝑓 𝑓 PROPN cana-3688 45 39 ( ( PUNCT cana-3688 45 40 π‘₯)𝑑π‘₯ π‘₯)𝑑π‘₯ PROPN cana-3688 45 41 𝑀 𝑀 PROPN cana-3688 45 42 . . PUNCT cana-3688 46 1 ( ( PUNCT cana-3688 46 2 𝑑 𝑑 X cana-3688 46 3 ) ) PUNCT cana-3688 46 4 = = VERB cana-3688 47 1 𝛼 𝛼 NOUN cana-3688 47 2 π‘₯ π‘₯ X cana-3688 47 3 𝑒 𝑒 PROPN cana-3688 47 4 𝑓 𝑓 DET cana-3688 47 5 ( ( PUNCT cana-3688 47 6 π‘₯ π‘₯ NOUN cana-3688 47 7 ) ) PUNCT cana-3688 47 8 𝑑π‘₯ 𝑑π‘₯ VERB cana-3688 47 9 + + NUM cana-3688 47 10 2 2 NUM cana-3688 47 11 𝛼 𝛼 NOUN cana-3688 47 12 ( ( PUNCT cana-3688 47 13 1 1 NUM cana-3688 47 14 βˆ’ βˆ’ NOUN cana-3688 47 15 𝛼 𝛼 NOUN cana-3688 47 16 ) ) PUNCT cana-3688 47 17 π‘₯ π‘₯ PROPN cana-3688 47 18 𝑒 𝑒 PROPN cana-3688 47 19 𝑓 𝑓 PROPN cana-3688 47 20 ( ( PUNCT cana-3688 47 21 π‘₯)𝑑π‘₯ π‘₯)𝑑π‘₯ PROPN cana-3688 47 22 + + CCONJ cana-3688 47 23 ( ( PUNCT cana-3688 47 24 1 1 NUM cana-3688 47 25 βˆ’ βˆ’ NOUN cana-3688 47 26 𝛼 𝛼 NOUN cana-3688 47 27 ) ) PUNCT cana-3688 47 28 𝑒 𝑒 PROPN cana-3688 47 29 𝑓 𝑓 PROPN cana-3688 47 30 ( ( PUNCT cana-3688 47 31 π‘₯)𝑑π‘₯ π‘₯)𝑑π‘₯ PROPN cana-3688 47 32 𝑀 𝑀 PROPN cana-3688 47 33 . . PUNCT cana-3688 48 1 ( ( PUNCT cana-3688 48 2 0 0 X cana-3688 48 3 ) ) PUNCT cana-3688 48 4 = = PRON cana-3688 49 1 𝛼 𝛼 NOUN cana-3688 49 2 π‘₯ π‘₯ X cana-3688 49 3 𝑒 𝑒 PROPN cana-3688 49 4 𝑓 𝑓 DET cana-3688 49 5 ( ( PUNCT cana-3688 49 6 π‘₯ π‘₯ NOUN cana-3688 49 7 ) ) PUNCT cana-3688 49 8 𝑑π‘₯ 𝑑π‘₯ VERB cana-3688 49 9 + + NUM cana-3688 49 10 2 2 NUM cana-3688 49 11 𝛼 𝛼 NOUN cana-3688 49 12 ( ( PUNCT cana-3688 49 13 1 1 NUM cana-3688 49 14 βˆ’ βˆ’ NOUN cana-3688 49 15 𝛼 𝛼 NOUN cana-3688 49 16 ) ) PUNCT cana-3688 49 17 π‘₯ π‘₯ PROPN cana-3688 49 18 𝑒 𝑒 PROPN cana-3688 49 19 𝑓 𝑓 PROPN cana-3688 49 20 ( ( PUNCT cana-3688 49 21 π‘₯)𝑑π‘₯ π‘₯)𝑑π‘₯ PROPN cana-3688 49 22 + + CCONJ cana-3688 49 23 ( ( PUNCT cana-3688 49 24 1 1 NUM cana-3688 49 25 βˆ’ βˆ’ NOUN cana-3688 49 26 𝛼 𝛼 NOUN cana-3688 49 27 ) ) PUNCT cana-3688 49 28 𝑒 𝑒 PROPN cana-3688 49 29 𝑓 𝑓 PROPN cana-3688 49 30 ( ( PUNCT cana-3688 49 31 π‘₯)𝑑π‘₯ π‘₯)𝑑π‘₯ PROPN cana-3688 49 32 𝑀 𝑀 PROPN cana-3688 49 33 . . PUNCT cana-3688 50 1 ( ( PUNCT cana-3688 50 2 0 0 X cana-3688 50 3 ) ) PUNCT cana-3688 50 4 = = PUNCT cana-3688 50 5 𝛼 𝛼 NOUN cana-3688 50 6 𝐸(𝑋 𝐸(𝑋 NOUN cana-3688 50 7 ) ) PUNCT cana-3688 51 1 + + CCONJ cana-3688 51 2 2 2 NUM cana-3688 51 3 𝛼 𝛼 NOUN cana-3688 51 4 ( ( PUNCT cana-3688 51 5 1 1 NUM cana-3688 51 6 βˆ’ βˆ’ PROPN cana-3688 51 7 𝛼)𝐸(𝑋 𝛼)𝐸(𝑋 NOUN cana-3688 51 8 ) ) PUNCT cana-3688 51 9 + + CCONJ cana-3688 51 10 ( ( PUNCT cana-3688 51 11 1 1 NUM cana-3688 51 12 βˆ’ βˆ’ NOUN cana-3688 51 13 𝛼 𝛼 NOUN cana-3688 51 14 ) ) PUNCT cana-3688 51 15 ( ( PUNCT cana-3688 51 16 12 12 NUM cana-3688 51 17 ) ) PUNCT cana-3688 51 18 ( ( PUNCT cana-3688 51 19 𝜎 𝜎 X cana-3688 51 20 ) ) PUNCT cana-3688 51 21 = = PUNCT cana-3688 51 22 𝛼 𝛼 NOUN cana-3688 51 23 𝐸(𝑋 𝐸(𝑋 NOUN cana-3688 51 24 ) ) PUNCT cana-3688 52 1 + + CCONJ cana-3688 52 2 2 2 NUM cana-3688 52 3 𝛼 𝛼 NOUN cana-3688 52 4 ( ( PUNCT cana-3688 52 5 1 1 NUM cana-3688 52 6 βˆ’ βˆ’ PROPN cana-3688 52 7 𝛼)𝐸(𝑋 𝛼)𝐸(𝑋 NOUN cana-3688 52 8 ) ) PUNCT cana-3688 52 9 + + CCONJ cana-3688 52 10 ( ( PUNCT cana-3688 52 11 1 1 NUM cana-3688 52 12 βˆ’ βˆ’ NOUN cana-3688 52 13 𝛼 𝛼 NOUN cana-3688 52 14 ) ) PUNCT cana-3688 52 15 βˆ’ βˆ’ PRON cana-3688 52 16 𝛼 𝛼 X cana-3688 52 17 𝐸(𝑋 𝐸(𝑋 NOUN cana-3688 52 18 ) ) PUNCT cana-3688 52 19 βˆ’ βˆ’ PROPN cana-3688 53 1 ( ( PUNCT cana-3688 53 2 1 1 NUM cana-3688 53 3 βˆ’ βˆ’ NUM cana-3688 53 4 𝛼 𝛼 NOUN cana-3688 53 5 ) ) PUNCT cana-3688 53 6 βˆ’ βˆ’ PROPN cana-3688 53 7 2 2 NUM cana-3688 53 8 𝛼 𝛼 NOUN cana-3688 53 9 ( ( PUNCT cana-3688 53 10 1 1 NUM cana-3688 53 11 βˆ’ βˆ’ NOUN cana-3688 53 12 𝛼 𝛼 NOUN cana-3688 53 13 ) ) PUNCT cana-3688 53 14 𝐸(𝑋 𝐸(𝑋 NOUN cana-3688 53 15 ) ) PUNCT cana-3688 53 16 ( ( PUNCT cana-3688 53 17 𝜎 𝜎 NOUN cana-3688 53 18 ) ) PUNCT cana-3688 53 19 = = PUNCT cana-3688 54 1 𝛼 𝛼 X cana-3688 54 2 𝜎 𝜎 NOUN cana-3688 54 3 𝜎 𝜎 NOUN cana-3688 54 4 = = X cana-3688 54 5 𝛼 𝛼 X cana-3688 54 6 𝜎 𝜎 PROPN cana-3688 54 7 ( ( PUNCT cana-3688 54 8 13 13 NUM cana-3688 54 9 ) ) PUNCT cana-3688 54 10 we we PRON cana-3688 54 11 get get VERB cana-3688 54 12 a a DET cana-3688 54 13 linear linear ADJ cana-3688 54 14 relation relation NOUN cana-3688 54 15 between between ADP cana-3688 54 16 deformable deformable ADJ cana-3688 54 17 standard standard ADJ cana-3688 54 18 deviation deviation NOUN cana-3688 54 19 and and CCONJ cana-3688 54 20 an an DET cana-3688 54 21 ordinary ordinary ADJ cana-3688 54 22 standard standard ADJ cana-3688 54 23 deviation deviation NOUN cana-3688 54 24 . . PUNCT cana-3688 55 1 deformable deformable ADJ cana-3688 55 2 standard standard ADJ cana-3688 55 3 deviation deviation NOUN cana-3688 55 4 is be AUX cana-3688 55 5 𝛼-times 𝛼-time NOUN cana-3688 55 6 the the DET cana-3688 55 7 ordinary ordinary ADJ cana-3688 55 8 standard standard ADJ cana-3688 55 9 deviation deviation NOUN cana-3688 55 10 . . PUNCT cana-3688 56 1 for for ADP cana-3688 56 2 𝛼 𝛼 NOUN cana-3688 56 3 = = SYM cana-3688 56 4 0 0 NUM cana-3688 56 5 , , PUNCT cana-3688 56 6 deformable deformable ADJ cana-3688 56 7 standard standard ADJ cana-3688 56 8 deviation deviation NOUN cana-3688 56 9 is be AUX cana-3688 56 10 zero zero NUM cana-3688 56 11 . . PUNCT cana-3688 57 1 as as SCONJ cana-3688 57 2 fractional fractional ADJ cana-3688 57 3 order order NOUN cana-3688 57 4 increases increase VERB cana-3688 57 5 deformable deformable ADJ cana-3688 57 6 standard standard ADJ cana-3688 57 7 deviation deviation NOUN cana-3688 57 8 also also ADV cana-3688 57 9 increase increase VERB cana-3688 57 10 . . PUNCT cana-3688 58 1 𝜎 𝜎 PROPN cana-3688 58 2 ≀ ≀ PROPN cana-3688 58 3 𝜎 𝜎 PROPN cana-3688 58 4 ( ( PUNCT cana-3688 58 5 14 14 NUM cana-3688 58 6 ) ) PUNCT cana-3688 58 7 example example NOUN cana-3688 58 8 1 1 NUM cana-3688 58 9 : : PUNCT cana-3688 58 10 for for ADP cana-3688 58 11 a a DET cana-3688 58 12 continuous continuous ADJ cana-3688 58 13 random random ADJ cana-3688 58 14 variable variable NOUN cana-3688 58 15 𝑋 𝑋 PROPN cana-3688 58 16 , , PUNCT cana-3688 58 17 probability probability NOUN cana-3688 58 18 density density NOUN cana-3688 58 19 function function NOUN cana-3688 58 20 of of ADP cana-3688 58 21 normal normal ADJ cana-3688 58 22 distribution distribution NOUN cana-3688 58 23 is be AUX cana-3688 58 24 defined define VERB cana-3688 58 25 as as ADP cana-3688 58 26 𝑓 𝑓 PROPN cana-3688 58 27 ( ( PUNCT cana-3688 58 28 π‘₯ π‘₯ NOUN cana-3688 58 29 ) ) PUNCT cana-3688 58 30 = = SYM cana-3688 58 31 √ √ PROPN cana-3688 58 32 𝑒 𝑒 PROPN cana-3688 58 33 ( ( PUNCT cana-3688 58 34 15 15 NUM cana-3688 58 35 ) ) PUNCT cana-3688 58 36 where where SCONJ cana-3688 58 37 βˆ’βˆž βˆ’βˆž ADP cana-3688 58 38 < < X cana-3688 58 39 π‘₯ π‘₯ X cana-3688 58 40 < < X cana-3688 58 41 ∞ ∞ PROPN cana-3688 58 42 and and CCONJ cana-3688 58 43 , , PUNCT cana-3688 58 44 βˆ’βˆž βˆ’βˆž X cana-3688 58 45 < < X cana-3688 58 46 πœ‡ πœ‡ ADP cana-3688 58 47 < < X cana-3688 58 48 ∞ ∞ PROPN cana-3688 58 49 , , PUNCT cana-3688 58 50 𝜎 𝜎 PROPN cana-3688 58 51 > > X cana-3688 58 52 0 0 PUNCT cana-3688 59 1 [ [ X cana-3688 59 2 3 3 NUM cana-3688 59 3 ] ] PUNCT cana-3688 59 4 , , PUNCT cana-3688 59 5 [ [ X cana-3688 59 6 5 5 NUM cana-3688 59 7 ] ] PUNCT cana-3688 59 8 . . PUNCT cana-3688 60 1 https://internationalpubls.com https://internationalpubls.com X cana-3688 60 2 441 441 NUM cana-3688 60 3 communications communication NOUN cana-3688 60 4 on on ADP cana-3688 60 5 applied apply VERB cana-3688 60 6 nonlinear nonlinear ADJ cana-3688 60 7 analysis analysis NOUN cana-3688 60 8 issn issn NOUN cana-3688 60 9 : : PUNCT cana-3688 60 10 1074 1074 NUM cana-3688 60 11 - - PUNCT cana-3688 60 12 133x 133x NUM cana-3688 60 13 vol vol NOUN cana-3688 60 14 . . PROPN cana-3688 61 1 32 32 NUM cana-3688 61 2 no no INTJ cana-3688 61 3 . . PUNCT cana-3688 62 1 8s 8s PROPN cana-3688 62 2 ( ( PUNCT cana-3688 62 3 2025 2025 NUM cana-3688 62 4 ) ) PUNCT cana-3688 62 5 on on ADP cana-3688 62 6 the the DET cana-3688 62 7 basis basis NOUN cana-3688 62 8 of of ADP cana-3688 62 9 equations equation NOUN cana-3688 62 10 ( ( PUNCT cana-3688 62 11 6 6 NUM cana-3688 62 12 ) ) PUNCT cana-3688 62 13 and and CCONJ cana-3688 62 14 ( ( PUNCT cana-3688 62 15 13 13 NUM cana-3688 62 16 ) ) PUNCT cana-3688 62 17 calculated calculate VERB cana-3688 62 18 value value NOUN cana-3688 62 19 of of ADP cana-3688 62 20 deformable deformable ADJ cana-3688 62 21 mean mean NOUN cana-3688 62 22 and and CCONJ cana-3688 62 23 deformable deformable ADJ cana-3688 62 24 standard standard ADJ cana-3688 62 25 deviation deviation NOUN cana-3688 62 26 given give VERB cana-3688 62 27 in in ADP cana-3688 62 28 the the DET cana-3688 62 29 following follow VERB cana-3688 62 30 table table NOUN cana-3688 62 31 , , PUNCT cana-3688 62 32 for for ADP cana-3688 62 33 mean mean NOUN cana-3688 62 34 πœ‡ πœ‡ ADP cana-3688 62 35 = = NOUN cana-3688 62 36 50 50 NUM cana-3688 62 37 and and CCONJ cana-3688 62 38 standard standard ADJ cana-3688 62 39 deviation deviation NOUN cana-3688 62 40 𝜎 𝜎 NOUN cana-3688 62 41 = = SYM cana-3688 62 42 25.3 25.3 NUM cana-3688 62 43 and and CCONJ cana-3688 62 44 fractional fractional ADJ cana-3688 62 45 order order NOUN cana-3688 62 46 𝛼 𝛼 NOUN cana-3688 62 47 = = SYM cana-3688 62 48 0.3 0.3 NUM cana-3688 62 49 , , PUNCT cana-3688 62 50 0.5 0.5 NUM cana-3688 62 51 , , PUNCT cana-3688 62 52 0.7 0.7 NUM cana-3688 62 53 , , PUNCT cana-3688 62 54 0.9 0.9 NUM cana-3688 62 55 , , PUNCT cana-3688 62 56 1 1 NUM cana-3688 62 57 . . PUNCT cana-3688 62 58 fractional fractional ADJ cana-3688 62 59 order order NOUN cana-3688 62 60 ( ( PUNCT cana-3688 62 61 𝜢 𝜢 NOUN cana-3688 62 62 ) ) PUNCT cana-3688 62 63 deformable deformable ADJ cana-3688 62 64 mean mean NOUN cana-3688 62 65 ( ( PUNCT cana-3688 62 66 𝝁𝜢 𝝁𝜢 NOUN cana-3688 62 67 ) ) PUNCT cana-3688 62 68 deformable deformable ADJ cana-3688 62 69 s.d s.d PROPN cana-3688 62 70 . . PUNCT cana-3688 63 1 ( ( PUNCT cana-3688 63 2 πˆπ‘Ώ πˆπ‘Ώ VERB cana-3688 63 3 𝜢 𝜢 PRON cana-3688 63 4 ) ) PUNCT cana-3688 63 5 0.3 0.3 NUM cana-3688 63 6 15.7 15.7 NUM cana-3688 63 7 7.59 7.59 NUM cana-3688 63 8 0.5 0.5 NUM cana-3688 63 9 25.5 25.5 NUM cana-3688 63 10 12.65 12.65 NUM cana-3688 63 11 0.7 0.7 NUM cana-3688 63 12 35.3 35.3 NUM cana-3688 63 13 17.71 17.71 NUM cana-3688 63 14 0.9 0.9 NUM cana-3688 63 15 45.5 45.5 NUM cana-3688 63 16 22.77 22.77 NUM cana-3688 63 17 1 1 NUM cana-3688 63 18 50 50 NUM cana-3688 63 19 25.3 25.3 NUM cana-3688 63 20 figure figure NOUN cana-3688 63 21 : : PUNCT cana-3688 63 22 normal normal ADJ cana-3688 63 23 distribution distribution NOUN cana-3688 63 24 plot plot NOUN cana-3688 63 25 of of ADP cana-3688 63 26 fractional fractional ADJ cana-3688 63 27 order order NOUN cana-3688 63 28 𝛼 𝛼 NOUN cana-3688 63 29 = = SYM cana-3688 63 30 0.3 0.3 NUM cana-3688 63 31 , , PUNCT cana-3688 63 32 0.5 0.5 NUM cana-3688 63 33 , , PUNCT cana-3688 63 34 0.7 0.7 NUM cana-3688 63 35 , , PUNCT cana-3688 63 36 0.9 0.9 NUM cana-3688 63 37 , , PUNCT cana-3688 63 38 1 1 NUM cana-3688 63 39 . . NOUN cana-3688 63 40 4 4 NUM cana-3688 63 41 . . X cana-3688 63 42 conclusion conclusion NOUN cana-3688 63 43 deformable deformable ADJ cana-3688 63 44 and and CCONJ cana-3688 63 45 ordinary ordinary ADJ cana-3688 63 46 mean mean NOUN cana-3688 63 47 have have VERB cana-3688 63 48 a a DET cana-3688 63 49 linear linear ADJ cana-3688 63 50 relationship relationship NOUN cana-3688 63 51 and and CCONJ cana-3688 63 52 deformable deformable ADJ cana-3688 63 53 standard standard ADJ cana-3688 63 54 deviation deviation NOUN cana-3688 63 55 is be AUX cana-3688 63 56 less less ADJ cana-3688 63 57 than than ADP cana-3688 63 58 and and CCONJ cana-3688 63 59 equal equal ADJ cana-3688 63 60 to to ADP cana-3688 63 61 ordinary ordinary ADJ cana-3688 63 62 standard standard ADJ cana-3688 63 63 deviation deviation NOUN cana-3688 63 64 . . PUNCT cana-3688 64 1 therefore therefore ADV cana-3688 64 2 , , PUNCT cana-3688 64 3 deformable deformable ADJ cana-3688 64 4 standard standard ADJ cana-3688 64 5 deviation deviation NOUN cana-3688 64 6 is be AUX cana-3688 64 7 more more ADV cana-3688 64 8 reliable reliable ADJ cana-3688 64 9 as as ADP cana-3688 64 10 compare compare NOUN cana-3688 64 11 to to ADP cana-3688 64 12 ordinary ordinary ADJ cana-3688 64 13 standard standard ADJ cana-3688 64 14 deviation deviation NOUN cana-3688 64 15 . . PUNCT cana-3688 65 1 deformable deformable ADJ cana-3688 65 2 mean mean NOUN cana-3688 65 3 along along ADP cana-3688 65 4 with with ADP cana-3688 65 5 deformable deformable ADJ cana-3688 65 6 standard standard ADJ cana-3688 65 7 deviation deviation NOUN cana-3688 65 8 gives give VERB cana-3688 65 9 more more ADJ cana-3688 65 10 insight insight NOUN cana-3688 65 11 information information NOUN cana-3688 65 12 of of ADP cana-3688 65 13 a a DET cana-3688 65 14 normal normal ADJ cana-3688 65 15 distribution distribution NOUN cana-3688 65 16 . . PUNCT cana-3688 66 1 we we PRON cana-3688 66 2 can can AUX cana-3688 66 3 see see VERB cana-3688 66 4 that that DET cana-3688 66 5 deformable deformable ADJ cana-3688 66 6 derivative derivative ADJ cana-3688 66 7 base base NOUN cana-3688 66 8 mean mean NOUN cana-3688 66 9 and and CCONJ cana-3688 66 10 standard standard ADJ cana-3688 66 11 deviation deviation NOUN cana-3688 66 12 affects affect VERB cana-3688 66 13 the the DET cana-3688 66 14 sharpness sharpness NOUN cana-3688 66 15 and and CCONJ cana-3688 66 16 flatness flatness NOUN cana-3688 66 17 of of ADP cana-3688 66 18 a a DET cana-3688 66 19 given give VERB cana-3688 66 20 normal normal ADJ cana-3688 66 21 distribution distribution NOUN cana-3688 66 22 , , PUNCT cana-3688 66 23 which which PRON cana-3688 66 24 shows show VERB cana-3688 66 25 the the DET cana-3688 66 26 memory memory NOUN cana-3688 66 27 of of ADP cana-3688 66 28 a a DET cana-3688 66 29 normal normal ADJ cana-3688 66 30 distribution distribution NOUN cana-3688 66 31 over over ADP cana-3688 66 32 varying vary VERB cana-3688 66 33 fractional fractional ADJ cana-3688 66 34 order order NOUN cana-3688 66 35 . . PUNCT cana-3688 67 1 therefore therefore ADV cana-3688 67 2 , , PUNCT cana-3688 67 3 we we PRON cana-3688 67 4 can can AUX cana-3688 67 5 say say VERB cana-3688 67 6 that that DET cana-3688 67 7 moment moment NOUN cana-3688 67 8 generating generate VERB cana-3688 67 9 base base NOUN cana-3688 67 10 deformable deformable ADJ cana-3688 67 11 means mean NOUN cana-3688 67 12 and and CCONJ cana-3688 67 13 deformable deformable ADJ cana-3688 67 14 standard standard ADJ cana-3688 67 15 deviation deviation NOUN cana-3688 67 16 is be AUX cana-3688 67 17 extension extension NOUN cana-3688 67 18 or or CCONJ cana-3688 67 19 ordinary ordinary ADJ cana-3688 67 20 mean mean NOUN cana-3688 67 21 and and CCONJ cana-3688 67 22 standard standard ADJ cana-3688 67 23 deviation deviation NOUN cana-3688 67 24 . . PUNCT cana-3688 68 1 further far ADV cana-3688 68 2 we we PRON cana-3688 68 3 can can AUX cana-3688 68 4 also also ADV cana-3688 68 5 extend extend VERB cana-3688 68 6 this this DET cana-3688 68 7 result result NOUN cana-3688 68 8 to to ADP cana-3688 68 9 other other ADJ cana-3688 68 10 measure measure NOUN cana-3688 68 11 of of ADP cana-3688 68 12 dispersion dispersion NOUN cana-3688 68 13 . . PUNCT cana-3688 69 1 references reference NOUN cana-3688 69 2 1 1 NUM cana-3688 69 3 . . NUM cana-3688 69 4 fahed fahe VERB cana-3688 69 5 zulfeqar zulfeqar PROPN cana-3688 69 6 , , PUNCT cana-3688 69 7 amit amit PROPN cana-3688 69 8 ujlayan ujlayan PROPN cana-3688 69 9 , , PUNCT cana-3688 69 10 priyanka priyanka PROPN cana-3688 69 11 ahuja ahuja PROPN cana-3688 69 12 , , PUNCT cana-3688 69 13 a a DET cana-3688 69 14 new new ADJ cana-3688 69 15 fractional fractional ADJ cana-3688 69 16 derivative derivative NOUN cana-3688 69 17 and and CCONJ cana-3688 69 18 its its PRON cana-3688 69 19 fractional fractional ADJ cana-3688 69 20 integral integral ADJ cana-3688 69 21 with with ADP cana-3688 69 22 some some DET cana-3688 69 23 applications application NOUN cana-3688 69 24 , , PUNCT cana-3688 69 25 xiv xiv PROPN cana-3688 69 26 : : PUNCT cana-3688 69 27 1705 1705 NUM cana-3688 69 28 . . PUNCT cana-3688 70 1 00962v1 00962v1 PUNCT cana-3688 71 1 [ [ X cana-3688 71 2 math.ca math.ca X cana-3688 71 3 ] ] X cana-3688 71 4 , , PUNCT cana-3688 71 5 2017 2017 NUM cana-3688 71 6 . . PUNCT cana-3688 72 1 https://internationalpubls.com https://internationalpubls.com X cana-3688 72 2 442 442 NUM cana-3688 72 3 communications communication NOUN cana-3688 72 4 on on ADP cana-3688 72 5 applied apply VERB cana-3688 72 6 nonlinear nonlinear ADJ cana-3688 72 7 analysis analysis NOUN cana-3688 72 8 issn issn NOUN cana-3688 72 9 : : PUNCT cana-3688 72 10 1074 1074 NUM cana-3688 72 11 - - PUNCT cana-3688 72 12 133x 133x NUM cana-3688 72 13 vol vol NOUN cana-3688 72 14 . . PROPN cana-3688 73 1 32 32 NUM cana-3688 73 2 no no INTJ cana-3688 73 3 . . PUNCT cana-3688 74 1 8s 8s PROPN cana-3688 74 2 ( ( PUNCT cana-3688 74 3 2025 2025 NUM cana-3688 74 4 ) ) PUNCT cana-3688 74 5 2 2 NUM cana-3688 74 6 . . PUNCT cana-3688 74 7 r. r. PROPN cana-3688 74 8 khalil khalil PROPN cana-3688 74 9 , , PUNCT cana-3688 74 10 m. m. NOUN cana-3688 74 11 al al PROPN cana-3688 74 12 horan horan PROPN cana-3688 74 13 , , PUNCT cana-3688 74 14 a. a. PROPN cana-3688 74 15 yousef yousef PROPN cana-3688 74 16 , , PUNCT cana-3688 74 17 & & CCONJ cana-3688 74 18 m. m. PROPN cana-3688 74 19 sababheh sababheh PROPN cana-3688 74 20 , , PUNCT cana-3688 74 21 a a DET cana-3688 74 22 new new ADJ cana-3688 74 23 definitions definition NOUN cana-3688 74 24 of of ADP cana-3688 74 25 fractional fractional ADJ cana-3688 74 26 derivative derivative NOUN cana-3688 74 27 . . PUNCT cana-3688 75 1 elsevier elsevier PROPN cana-3688 75 2 , , PUNCT cana-3688 75 3 journal journal NOUN cana-3688 75 4 of of ADP cana-3688 75 5 computational computational ADJ cana-3688 75 6 and and CCONJ cana-3688 75 7 applied applied ADJ cana-3688 75 8 mathematics mathematic NOUN cana-3688 75 9 , , PUNCT cana-3688 75 10 2014 2014 NUM cana-3688 75 11 , , PUNCT cana-3688 75 12 264 264 NUM cana-3688 75 13 , , PUNCT cana-3688 75 14 65 65 NUM cana-3688 75 15 - - SYM cana-3688 75 16 70 70 NUM cana-3688 75 17 . . PUNCT cana-3688 76 1 3 3 X cana-3688 76 2 . . X cana-3688 76 3 miller miller PROPN cana-3688 76 4 , , PUNCT cana-3688 76 5 irvin irvin PROPN cana-3688 76 6 & & CCONJ cana-3688 76 7 miller miller PROPN cana-3688 76 8 , , PUNCT cana-3688 76 9 marylee marylee PROPN cana-3688 76 10 ’s ’s PART cana-3688 76 11 john john PROPN cana-3688 76 12 e. e. PROPN cana-3688 76 13 freund freund PROPN cana-3688 76 14 ’s ’s PART cana-3688 76 15 , , PUNCT cana-3688 76 16 mathematical mathematical ADJ cana-3688 76 17 statistics statistic NOUN cana-3688 76 18 with with ADP cana-3688 76 19 applications application NOUN cana-3688 76 20 ( ( PUNCT cana-3688 76 21 7th 7th ADJ cana-3688 76 22 edition edition NOUN cana-3688 76 23 ) ) PUNCT cana-3688 76 24 . . PUNCT cana-3688 77 1 pearson pearson PROPN cana-3688 77 2 education education PROPN cana-3688 77 3 asia asia PROPN cana-3688 77 4 , , PUNCT cana-3688 77 5 2006 2006 NUM cana-3688 77 6 . . PUNCT cana-3688 78 1 4 4 X cana-3688 78 2 . . X cana-3688 78 3 miller miller PROPN cana-3688 78 4 , , PUNCT cana-3688 78 5 irvin irvin PROPN cana-3688 78 6 & & CCONJ cana-3688 78 7 miller miller PROPN cana-3688 78 8 , , PUNCT cana-3688 78 9 marylee marylee PROPN cana-3688 78 10 ’s ’s PART cana-3688 78 11 john john PROPN cana-3688 78 12 e. e. PROPN cana-3688 78 13 freund freund PROPN cana-3688 78 14 ’s ’s PART cana-3688 78 15 , , PUNCT cana-3688 78 16 mathematical mathematical ADJ cana-3688 78 17 statistics statistic NOUN cana-3688 78 18 with with ADP cana-3688 78 19 applications application NOUN cana-3688 78 20 ( ( PUNCT cana-3688 78 21 7th 7th ADJ cana-3688 78 22 edition edition NOUN cana-3688 78 23 ) ) PUNCT cana-3688 78 24 . . PUNCT cana-3688 79 1 pearson pearson PROPN cana-3688 79 2 education education PROPN cana-3688 79 3 asia asia PROPN cana-3688 79 4 , , PUNCT cana-3688 79 5 2006 2006 NUM cana-3688 79 6 . . PUNCT cana-3688 80 1 5 5 NUM cana-3688 80 2 . . X cana-3688 80 3 jay jay PROPN cana-3688 80 4 l. l. PROPN cana-3688 80 5 devore devore PROPN cana-3688 80 6 , , PUNCT cana-3688 80 7 & & CCONJ cana-3688 80 8 kenneth kenneth PROPN cana-3688 80 9 n. n. PROPN cana-3688 80 10 berk berk PROPN cana-3688 80 11 , , PUNCT cana-3688 80 12 mathematical mathematical ADJ cana-3688 80 13 statistics statistic NOUN cana-3688 80 14 with with ADP cana-3688 80 15 applications application NOUN cana-3688 80 16 , , PUNCT cana-3688 80 17 thomson thomson PROPN cana-3688 80 18 brooks brooks PROPN cana-3688 80 19 cole cole PROPN cana-3688 80 20 , , PUNCT cana-3688 80 21 2007 2007 NUM cana-3688 80 22 . . PUNCT cana-3688 81 1 6 6 NUM cana-3688 81 2 . . PUNCT cana-3688 81 3 robert robert PROPN cana-3688 81 4 v. v. PROPN cana-3688 81 5 hogg hogg PROPN cana-3688 81 6 , , PUNCT cana-3688 81 7 alen alen PROPN cana-3688 81 8 craig craig PROPN cana-3688 81 9 , , PUNCT cana-3688 81 10 joseph joseph PROPN cana-3688 81 11 w. w. PROPN cana-3688 81 12 mckean mckean PROPN cana-3688 81 13 , , PUNCT cana-3688 81 14 introduction introduction NOUN cana-3688 81 15 to to ADP cana-3688 81 16 mathematical mathematical ADJ cana-3688 81 17 statistics statistic NOUN cana-3688 81 18 , , PUNCT cana-3688 81 19 sixth sixth ADJ cana-3688 81 20 edition edition NOUN cana-3688 81 21 , , PUNCT cana-3688 81 22 pearson pearson PROPN cana-3688 81 23 , , PUNCT cana-3688 81 24 2011 2011 NUM cana-3688 81 25 . . PUNCT cana-3688 82 1 7 7 X cana-3688 82 2 . . X cana-3688 82 3 s.p s.p PROPN cana-3688 82 4 . . PUNCT cana-3688 82 5 gupta gupta PROPN cana-3688 82 6 , , PUNCT cana-3688 82 7 statistical statistical ADJ cana-3688 82 8 method method NOUN cana-3688 82 9 , , PUNCT cana-3688 82 10 sultan sultan PROPN cana-3688 82 11 chand chand PROPN cana-3688 82 12 & & CCONJ cana-3688 82 13 sons son NOUN cana-3688 82 14 , , PUNCT cana-3688 82 15 2013 2013 NUM cana-3688 82 16 . . PUNCT cana-3688 83 1 8 8 X cana-3688 83 2 . . PUNCT cana-3688 84 1 k.b k.b PROPN cana-3688 84 2 . . PROPN cana-3688 84 3 oldham oldham PROPN cana-3688 84 4 , , PUNCT cana-3688 84 5 & & CCONJ cana-3688 84 6 j. j. PROPN cana-3688 84 7 spanier spanier PROPN cana-3688 84 8 , , PUNCT cana-3688 84 9 the the DET cana-3688 84 10 fractional fractional ADJ cana-3688 84 11 calculus calculus NOUN cana-3688 84 12 , , PUNCT cana-3688 84 13 new new PROPN cana-3688 84 14 york york PROPN cana-3688 84 15 , , PUNCT cana-3688 84 16 academic academic ADJ cana-3688 84 17 press press NOUN cana-3688 84 18 , , PUNCT cana-3688 84 19 1974 1974 NUM cana-3688 84 20 . . PUNCT cana-3688 85 1 9 9 NUM cana-3688 85 2 . . X cana-3688 85 3 udita udita PROPN cana-3688 85 4 n. n. PROPN cana-3688 85 5 katugampola katugampola PROPN cana-3688 85 6 , , PUNCT cana-3688 85 7 new new ADJ cana-3688 85 8 fractional fractional ADJ cana-3688 85 9 derivative derivative NOUN cana-3688 85 10 with with ADP cana-3688 85 11 classical classical ADJ cana-3688 85 12 properties property NOUN cana-3688 85 13 , , PUNCT cana-3688 85 14 journal journal NOUN cana-3688 85 15 of of ADP cana-3688 85 16 the the DET cana-3688 85 17 american american PROPN cana-3688 85 18 mathematical mathematical PROPN cana-3688 85 19 society society NOUN cana-3688 85 20 , , PUNCT cana-3688 85 21 vol vol NOUN cana-3688 85 22 . . PROPN cana-3688 85 23 00 00 NUM cana-3688 85 24 , , PUNCT cana-3688 85 25 no no INTJ cana-3688 85 26 . . NOUN cana-3688 85 27 0 0 NUM cana-3688 85 28 , , PUNCT cana-3688 85 29 pp pp ADJ cana-3688 85 30 . . PUNCT cana-3688 85 31 000 000 NUM cana-3688 85 32 - - PUNCT cana-3688 85 33 000 000 NUM cana-3688 85 34 , , PUNCT cana-3688 85 35 arxiv:1410.6535v2 arxiv:1410.6535v2 CCONJ cana-3688 86 1 [ [ X cana-3688 86 2 math.ca math.ca X cana-3688 86 3 ] ] X cana-3688 86 4 8 8 NUM cana-3688 86 5 nov nov PROPN cana-3688 86 6 . . PROPN cana-3688 86 7 2014 2014 NUM cana-3688 86 8 . . PUNCT cana-3688 87 1 10 10 NUM cana-3688 87 2 . . PUNCT cana-3688 87 3 udita udita PROPN cana-3688 87 4 n. n. PROPN cana-3688 87 5 katugampola katugampola PROPN cana-3688 87 6 , , PUNCT cana-3688 87 7 a a DET cana-3688 87 8 new new ADJ cana-3688 87 9 fractional fractional ADJ cana-3688 87 10 derivative derivative NOUN cana-3688 87 11 with with ADP cana-3688 87 12 classical classical ADJ cana-3688 87 13 properties property NOUN cana-3688 87 14 , , PUNCT cana-3688 87 15 journal journal NOUN cana-3688 87 16 of of ADP cana-3688 87 17 american american PROPN cana-3688 87 18 mathematical mathematical PROPN cana-3688 87 19 society society NOUN cana-3688 87 20 volume volume NOUN cana-3688 87 21 , , PUNCT cana-3688 87 22 2014 2014 NUM cana-3688 87 23 . . PUNCT cana-3688 88 1 11 11 NUM cana-3688 88 2 . . X cana-3688 88 3 tarasov tarasov NOUN cana-3688 88 4 , , PUNCT cana-3688 88 5 vasily vasily PROPN cana-3688 88 6 e. e. PROPN cana-3688 88 7 , , PUNCT cana-3688 88 8 no no DET cana-3688 88 9 violation violation NOUN cana-3688 88 10 of of ADP cana-3688 88 11 the the DET cana-3688 88 12 leibniz leibniz PROPN cana-3688 88 13 rule rule NOUN cana-3688 88 14 , , PUNCT cana-3688 88 15 no no DET cana-3688 88 16 fractional fractional ADJ cana-3688 88 17 derivative derivative ADJ cana-3688 88 18 , , PUNCT cana-3688 88 19 elsevier elsevier NOUN cana-3688 88 20 , , PUNCT cana-3688 88 21 communication communication NOUN cana-3688 88 22 to to ADP cana-3688 88 23 nonlinear nonlinear ADJ cana-3688 88 24 science science NOUN cana-3688 88 25 and and CCONJ cana-3688 88 26 numerical numerical PROPN cana-3688 88 27 simulation simulation PROPN cana-3688 88 28 , , PUNCT cana-3688 88 29 2013 2013 NUM cana-3688 88 30 , , PUNCT cana-3688 88 31 18 18 NUM cana-3688 88 32 , , PUNCT cana-3688 88 33 2945 2945 NUM cana-3688 88 34 - - SYM cana-3688 88 35 2948 2948 NUM cana-3688 88 36 . . PUNCT cana-3688 89 1 12 12 NUM cana-3688 89 2 . . PUNCT cana-3688 89 3 tarasov tarasov NOUN cana-3688 89 4 , , PUNCT cana-3688 89 5 vasily vasily PROPN cana-3688 89 6 e. e. PROPN cana-3688 89 7 , , PUNCT cana-3688 89 8 on on ADP cana-3688 89 9 chain chain NOUN cana-3688 89 10 rule rule NOUN cana-3688 89 11 of of ADP cana-3688 89 12 fractional fractional ADJ cana-3688 89 13 derivatives derivative NOUN cana-3688 89 14 , , PUNCT cana-3688 89 15 elsevier elsevier NOUN cana-3688 89 16 , , PUNCT cana-3688 89 17 communication communication NOUN cana-3688 89 18 to to ADP cana-3688 89 19 nonlinear nonlinear ADJ cana-3688 89 20 science science NOUN cana-3688 89 21 and and CCONJ cana-3688 89 22 numerical numerical PROPN cana-3688 89 23 simulation simulation PROPN cana-3688 89 24 , , PUNCT cana-3688 89 25 30 30 NUM cana-3688 89 26 ( ( PUNCT cana-3688 89 27 2016 2016 NUM cana-3688 89 28 ) ) PUNCT cana-3688 89 29 , , PUNCT cana-3688 89 30 pp pp PROPN cana-3688 89 31 . . PUNCT cana-3688 90 1 1 1 NUM cana-3688 90 2 - - SYM cana-3688 90 3 4 4 NUM cana-3688 90 4 . . NOUN cana-3688 90 5 13 13 NUM cana-3688 90 6 . . PUNCT cana-3688 91 1 abdeljawad abdeljawad NOUN cana-3688 91 2 , , PUNCT cana-3688 91 3 t. t. PROPN cana-3688 91 4 , , PUNCT cana-3688 91 5 on on ADP cana-3688 91 6 conformable conformable ADJ cana-3688 91 7 fractional fractional ADJ cana-3688 91 8 calculus calculus NOUN cana-3688 91 9 , , PUNCT cana-3688 91 10 elsevier elsevier NOUN cana-3688 91 11 , , PUNCT cana-3688 91 12 journal journal NOUN cana-3688 91 13 of of ADP cana-3688 91 14 computational computational ADJ cana-3688 91 15 and and CCONJ cana-3688 91 16 applied applied ADJ cana-3688 91 17 mathematics mathematic NOUN cana-3688 91 18 , , PUNCT cana-3688 91 19 2015 2015 NUM cana-3688 91 20 , , PUNCT cana-3688 91 21 279 279 NUM cana-3688 91 22 , , PUNCT cana-3688 91 23 57 57 NUM cana-3688 91 24 - - SYM cana-3688 91 25 66 66 NUM cana-3688 91 26 . . PUNCT cana-3688 92 1 https://internationalpubls.com https://internationalpubls.com X cana-3688 92 2 443 443 NUM